Semimartingales on Duals of Nuclear Spaces

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Ustünel in ¨ -al [email protected] E-mail: ′ a o´,15126,CsaRica. Costa José, 11501-2060, San fancersaeΦt aeaΦ a have to Φ space nuclear a of ′ vle eiatnae sapoetv ytmo Hilbert of system projective a as semimartingales -valued yidia eiatnae,smmrigls ulo a of dual semimartingales, semimartingales, cylindrical .A Fonseca-Mora A. C. Introduction 1 s¨ nlue h atta ne hs assumptions these under that fact the Ustünel used ¨ Abstract ′ t togda.Udrteasmto htΦ that assumption the Under dual. strong its ′ vle Lévy process. -valued 1 ′ vle eiatnae a rtyintro- firstly was semimartingales -valued 01,6G7 02,60G48. 60G20, 60G17, 60B11, ′ vle eiatnaeversion semimartingale -valued sr n ucetconditions sufficient and ssary ′ fsmlrntr u o more for but nature similar of vle eiatnae and semimartingales -valued 7 9 0 8) u othe to Due 38]). 30, 29, 27, , oia ersnain Lévy representation, nonical igfrefrteetpsof types these for force ving saeas rvdd Later, provided. also are es td fclnrclstochas- cylindrical of study s¨ nli i subsequent his Ustünel in hsi acls ti only is it calculus, chastic ¨ stediigniet a to noise driving the as ecneto predictable of concept he oiain steueof use the is motivations tnaetkn ausin values taking rtingale nrclsemimartingale indrical ibr-cmd em- Hilbert-Schmidt lo general a of al ′ works [34, 35, 36], by Pérez-Abreu [24], and by other authors (e.g. [3, 6, 25]). How- ever, Ustünel’s¨ approach for semimartingales as projective systems can no longer be applied if one assumes that Φ is a general nuclear space, because in that case Φ′ is usually not nuclear nor complete. Therefore, if Φ is assumed only to be nuclear, a new theory of semimartingales has to be developed and this is exactly the purpose of this paper; to begin a systematic study of cylindrical semimartingales and semimartingales in the strong dual Φ′ of a general nuclear space Φ. Our main motivation is the further introduction of a theory of stochastic integration and of stochastic partial differential equations driven by semimartingales in duals of nuclear spaces. This program was carried out by the author in [11] for the Lévy case, but for the semimartingale case a new approach has to be considered. The necessary properties of semimartingales are developed in this paper. The corresponding application to stochastic integration and existence of solutions to stochastic partial differential equations will appear in forthcoming papers. We start in Sect. 2 by setting our notation and by reviewing some useful properties of nuclear spaces, cylindrical processes, and the space S0 of real-valued semimartingales. Then, in Sect. 3 we address the first main problem of our study; it consists in to establish conditions under which a cylindrical semimartingale X = (X : t 0) t ≥ in Φ′ does have a semimartingale version with càdlàg (i.e. right-continuous with left limits) paths. In the literature this procedure is often known as “regularization” (see [10, 13, 24]). Our main result is Theorem 3.7 where it is shown that a sufficient condition is equicontinuity for each T > 0 of the family (X : t [0, T ]) as operators from Φ t ∈ into the space of real-valued random variables L0 (Ω, F , P) defined on a given probability space (Ω, F , P). In Proposition 3.14 we show that this condition is equivalent to the condition that X as a linear map from Φ into S0 be continuous. As a consequence of our result we show that if the nuclear space is ultrabornological (e.g. Fréchet), then each Φ′-valued semimartingale with Radon laws has a càdlàg version. Regular- ization theorems for more specific classes of semimartingales and examples are also considered. Our regularization results generalize those obtained by Ustünel¨ in [33, 34] and by Pérez-Abreu [24] where only Φ′-valued semimartingales (but not cylindrical semimartingales) were considered and it is assumed that Φ′ is complete nuclear. Our second main aim on this work is to find a canonical representation for Φ′- valued semimartingales. This is done in Sect. 4 by using our regularization results from Sect. 3. Our main result is Theorem 4.2 where we show that if for a Φ′-valued semimartingale we assume equicontinuity for each T > 0 of the induced cylindrical process (X : t [0, T ]), then X possesses a canonical representation similar to that t ∈ for semimartingales in finite dimensions. This equicontinuity assumption is always satisfied if the nuclear space is ultrabornological and the semimartingale have Radon laws. Furthermore, we introduce the concept of predictable characteristics for Φ′- valued semimartingales. To the extend of our knowledge the only previous work that considered the existence of a canonical representation for semimartingales in the dual of a nuclear Fréchet space was carried out by Pérez-Abreu in [24]. Observe that our result generalize that of Pérez-Abreu to semimartingales in the dual of a general nuclear space. Finally, in Sect. 5 we examine in detail the canonical representation of a Φ′-valued Lévy process and its relation with their Lévy-Itôdecomposition studied by the author in [12]. It is shown that the predictable characteristics of a Lévy process coincide with those of its Lévy-Khintchine formula and that the particular form of these character-

2 istics distinguish the L´evy process among the Φ′-valued semimartingales. We hope that our semimartingale canonical representation and our deﬁnition of characteristics could be used in the future to study further properties of Φ′-valued semimartingales, as for example to study functional central limit theorems.

2 Deﬁnitions and Notation

Let Φ be a locally convex space (we will only consider vector spaces over R). If p is a continuous semi-norm on Φ and r > 0, the closed ball of radius r of p given by B (r) = φ Φ : p(φ) r is a closed, convex, balanced neighborhood of zero p { ∈ ≤ } in Φ. A continuous semi-norm (respectively a norm) p on Φ is called Hilbertian if p(φ)2 = Q(φ, φ), for all φ Φ, where Q is a symmetric, non-negative bilinear form ∈ (respectively inner product) on Φ Φ. Let Φ be the Hilbert space that corresponds to × p the completion of the pre-Hilbert space (Φ/ker(p), p˜), wherep ˜(φ + ker(p)) = p(φ) for each φ Φ. The quotient map Φ Φ/ker(p) has an unique continuous linear extension ∈ → i :Φ Φ . Let q be another continuous Hilbertian semi-norm on Φ for which p → p p q. In this case, ker(q) ker(p). Moreover, the inclusion map from Φ/ker(q) into ≤ ⊆ Φ/ker(p) is linear and continuous, and therefore it has a unique continuous extension i :Φ Φ . Furthermore, we have the following relation: i = i i . p,q q → p p p,q ◦ q We denote by Φ′ the topological dual of Φ and by f , φ the canonical pairing of h i elements f Φ′, φ Φ. Unless otherwise specified, Φ′ will always be consider equipped ∈ ∈ with its strong topology, i.e. the topology on Φ′ generated by the family of semi-norms (ηB), where for each B Φ bounded we have ηB(f) = sup f , φ : φ B for all ′ ⊆ {|h i| ∈ } ′ f Φ . If p is a continuous Hilbertian semi-norm on Φ, then we denote by Φp the ∈ ′ ′ ′ Hilbert space dual to Φp. The dual norm p on Φp is given by p (f) = sup f , φ : ′ ′ {|h i| φ Bp(1) for all f Φp. Moreover, the dual operator ip corresponds to the canonical ∈ } ′ ∈ ′ inclusion from Φp into Φ and it is linear and continuous. Let p and q be continuous Hilbertian semi-norms on Φ such that p q. The space of ≤ continuous linear operators (respectively Hilbert-Schmidt operators) from Φq into Φp is denoted by (Φ , Φ ) (respectively (Φ , Φ )) and the operator norm (respectively L q p L2 q p Hilbert-Schmidt norm) is denote by L(Φq ,Φp) (respectively L2(Φq ,Φp)). We employ ||·|| ||·||′ ′ an analogous notation for operators between the dual spaces Φp and Φq. A locally convex space is called ultrabornological if it is the inductive limit of a family of Banach spaces. A barreled space is a locally convex space such that every convex, balanced, absorbing and closed subset is a neighborhood of zero. For equivalent definitions see [16, 23]. Let us recall that a (Hausdorff) locally convex space (Φ, ) is called nuclear if its T topology is generated by a family Π of Hilbertian semi-norms such that for each T p Π there exists q Π, satisfying p q and the canonical inclusion i :Φ Φ is ∈ ∈ ≤ p,q q → p Hilbert-Schmidt. Other equivalent definitions of nuclear spaces can be found in [26, 32]. Let Φ be a nuclear space. If p is a continuous Hilbertian semi-norm on Φ, then the Hilbert space Φp is separable (see [26], Proposition 4.4.9 and Theorem 4.4.10, p.82). Now, let (p : n N) be an increasing sequence of continuous Hilbertian semi-norms n ∈ on (Φ, ). We denote by θ the locally convex topology on Φ generated by the family T (p : n N). The topology θ is weaker than . We will call θ a (weaker) countably n ∈ T Hilbertian topology on Φ and we denote by Φθ the space (Φ,θ) and by Φθ its completion. The space Φθ is a (not necessarily Hausdorff) separable, complete, pseudo-metrizable (hence Baire and ultrabornological; see Example 13.2.8(b) and Theoremb 13.2.12 in b 3 ′ ′ ′ [23]) locally convex space and its dual space satisfies (Φθ) = (Φθ) = n∈N Φpn (see [10], Proposition 2.4). S b Example 2.1. It is well-known (see e.g. [26, 31, 32]) that the space of test functions E := ∞(K) (K: compact subset of Rd), E := ∞(Rd), the rapidly decreasing K C C functions S (Rd), and the space of harmonic functions (U) (U: open subset of Rd; H see [26], Section 6.3), are all examples of Fréchet nuclear spaces. Their (strong) dual E ′ E ′ S ′ Rd ′ spaces K , , ( ), (U), are also nuclear spaces. On the other hand, the space D H ∞ Rd of test functions (U) := c (U) (U: open subset of ), the space of polynomials C N in n-variables, the space of real-valued sequences R (with direct sum topology) Pn are strict inductive limits of Fréchet nuclear spaces (hence they are also nuclear). The space of distributions D ′(U) (U: open subset of Rd) is also nuclear. All the above are examples of (complete) ultrabornological nuclear spaces. Other interesting examples of nuclear spaces are the space of continuous linear operators between a semi-reflexive dual nuclear space into a nuclear space, and tensor products of arbitrary nuclear spaces. These spaces need not be ultrabornological as for example happens with the space D(R) S (Rd) (see [4]). ⊗ F P Throughoutb this work we assume that (Ω, , ) is a complete probability space and consider a filtration ( : t 0) on (Ω, F , P) that satisfies the usual conditions, i.e. Ft ≥ it is right continuous and contains all subsets of sets of of P-measure zero. We F0 F denote by L0 (Ω, F , P) the space of equivalence classes of real-valued random variables defined on (Ω, F , P). We always consider the space L0 (Ω, F , P) equipped with the topology of convergence in probability and in this case it is a complete, metrizable, topological vector space. A cylindrical random variable in Φ′ is a linear map X :Φ L0 (Ω, F , P) (see [10]). → If X is a cylindrical random variable in Φ′, we say that X is n-integrable (n N) if ∈ E ( X(φ) n) < , φ Φ, and has zero-mean if E (X(φ)) = 0, φ Φ. The Fourier | | ∞ ∀ ∈ ∀ ∈ transform of X is the map from Φ into C given by φ E(eiX(φ)). 7→ Let X beaΦ′-valued random variable, i.e. X :Ω Φ′ is a F / (Φ′ )-measurable → β B β map. For each φ Φ we denote by X , φ the real-valued random variable defined ∈ h i by X , φ (ω) := X(ω) , φ , for all ω Ω. The linear mapping φ X , φ is called h i h i ∈ 7→ h i the cylindrical random variable induced/defined by X. We will say that a Φ′-valued random variable X is n-integrable (n N) if the cylindrical random variable induced ∈ by X is n-integrable. Let J = R := [0, ) or J = [0, T ] for T > 0. We say that X = (X : t J) is a + ∞ t ∈ cylindrical process in Φ′ if X is a cylindrical random variable for each t J. Clearly, t ∈ any Φ′-valued stochastic processes X = (X : t J) induces/defines a cylindrical t ∈ process under the prescription: X , φ = ( X , φ : t J), for each φ Φ. h i h t i ∈ ∈ If X is a cylindrical random variable in Φ′,aΦ′-valued random variable Y is called a version of X if for every φ Φ, X(φ)= Y , φ P-a.e. A Φ′-valued process Y = (Y : ∈ h i t t J) is said to be a Φ′-valued version of the cylindrical process X = (X : t J) on ∈ t ∈ Φ′ if for each t J, Y is a Φ′-valued version of X . ∈ t t For a Φ′-valued process X = (X : t J) terms like continuous, càdlàg, purely t ∈ discontinuous, adapted, predictable, etc. have the usual (obvious) meaning. AΦ′-valued random variable X is called regular if there exists a weaker countably Hilbertian topology θ on Φ such that P(ω : X(ω) (Φ )′) = 1. Furthermore, a Φ′- ∈ θ valued process Y = (Y : t J) is said to be regular if Y is a regular random variable t ∈ t b

4 for each t J. In that case the law of each Y is a Radon measure in Φ′ (see Theorem ∈ t 2.10 in [10]). If Ψ is a separable Hilbert space, recall that a Ψ-valued adapted càdlàg process X = (X : t 0) is a semimartingale if it admit a representation of the form t ≥ X = X + M + A , t 0, t 0 t t ∀ ≥ where M = (M : t 0) is a càdlàg local martingale and A = (A : t 0) is a t ≥ t ≥ càdlàg adapted process of finite variation, and M0 = A0 = 0. The reader is referred to [21] for the basic theory of Hilbert space valued semimartingales and to [7, 15, 28] for Rd-valued semimartingales. We denote by S0 the linear space of (equivalence classes) of real-valued semimartingales. We consider on S0 the topology given in Memin [22]: define on S0 by |·|0 ∞ z = 2−k z , | |0 | |k Xk=1 where for each k N, ∈ E z k = sup (1 (h z)k ) , | | h∈E1 ∧ | · | is the class of real-valued predictable processes h of the form E1 n−1

h = ai ½(ti,ti+1]×Ω, Xi=1 for 0 < t < t < ...t < , a is an -measurable random variable, a 1, 1 2 n ∞ i Fti | i| ≤ i = 1,...,n 1, and − k n−1 (h z) = h dz = a z z . · k s s i ti+1∧k − ti∧k Z0 i=1 X Then, d(y, z)= y z is a metric on S0 and (S0, d) is a complete, metric, topological | − |0 vector space (is not in general locally convex). Unless otherwise speciﬁed, the space S0 will always be consider equipped with this topology.

If x = (x : t 0) is a real-valued semimartingale, we denote by x p (1 p< ) t ≥ || ||S ≤ ∞ the following quantity:

∞ 1/2 x Sp = inf [m,m]∞ + das : x = m + a , || || | | p ( Z0 L (Ω,F ,P) )

where the inﬁmum is taken over all the decompositions x = m + a as a sum of a local martingale m and a process of ﬁnite variation a. Recall that ([m,m] : t 0) denotes t ≥ the quadratic variation process associated to the local martingale m, i.e. [m,m]t = mc,mc + (∆m )2, where mc is the (unique) continuous local martingale part hh iit 0≤s≤t s of m and ( mc,mc : t 0) its angle bracket process (see Section I in [15]). The set hhP iit ≥ of all semimartingales x for which x p < is a Banach space under the norm p || ||S ∞ ||·||S and is denoted by Sp (see VII.98 in [7]). Furthermore, if x = m + a is a decomposition of x such that x p < it is known that in such a case a is of integrable variation || ||S ∞ (see VII.98(c) in [7]).

5 3 Cylindrical Semimartingales in the Dual of a Nuclear Space

Assumption 3.1. Throught this section and unless otherwise speciﬁed Φ will always denote a nuclear space.

3.1 Regularization of Cylindrical Semimartigales In this section our main objective is to establish sufficient conditions for a cylindrical semimartingale in Φ′ to have a Φ′-valued càdlàg semimartingale version. Following [10] and [24], we call such a result as regularization of cylindrical semimartingales (see Theorem 3.7). As a consequence of our results for cylindrical semimartingales we will show that if we assume some extra structure on the space Φ then every Φ′- semimartingale has a càdlàg version (see Proposition 3.12). We begin by introducing our definition of cylindrical semimartingales in duals of nuclear spaces.

′ Definition 3.2. A cylindrical semimartingale in Φ is a cylindrical process X = (Xt : t 0) in Φ′ such that φ Φ, X(φ) is a real-valued semimartingale. A cylindrical local ≥ ∀ ∈ martingale (resp. cylindrical martingale) in Φ′ is a cylindrical process M = (M : t 0) t ≥ for which M(φ) = (M (φ) : t 0) is a real-valued local martingale (resp. a martingale). t ≥ In a similar way, a cylindrical finite variation process in Φ′ is a cylindrical process A = (A : t 0) for which A(φ) = (A (φ) : t 0) is a real-valued process of finite t ≥ t ≥ variation (i.e. if P-a.a. of its paths have locally bounded variation).

If X is a cylindrical semimartingale in Φ′, then each X(φ) has the decomposition

Xφ = Xφ + M φ + Aφ, t 0, (3.1) t 0 t t ∀ ≥ φ φ φ where M = (Mt : t 0) is a real-valued càdlàg local martingale with M0 = 0, φ φ ≥ and A = (At : t 0) is a real-valued càdlàg adapted process of finite variation and φ ≥ φ A0 = 0. If there exists a decomposition (3.1) with each A predictable, we will say that X is a special cylindrical semimartingale. It is important to stress the fact that in general the maps φ M φ and φ Aφ do not define a cylindrical local martingale 7→ 7→ and a cylindrical finite variation process because they might be not linear operators. For more details see the discussion in Example 3.10 in [1] and Remark 2.2 in [17].

Definition 3.3. AΦ′-valued process X = (X : t 0) is a semimartingale if the t ≥ induced cylindrical process is a cylindrical semimartingale. In a completely analogue way we define the concepts of Φ′-valued special semimartingale, martingale, local martingale and process of finite variation.

Example 3.4. One important example of a cylindrical semimartingale in Φ′ is a cylindrical Lévy process, i.e. a cylindrical process L = (L : t 0) in Φ′ such that n N, t ≥ ∀ ∈ φ ,...,φ Φ, the Rn-valued process ((L (φ ),...,L (φ )) : t 0) is a Lévy process. 1 n ∈ t 1 t n ≥ In a similar way, an example of a Φ′-valued semimartingale is a Lévy processes. Recall that a Φ′-valued process L = (L : t 0) is called a Lévy process if (i) L = 0 t ≥ 0 a.s., (ii) L has independent increments, i.e. for any n N, 0 t

6 measure and the mapping t µ from R into the space of Radon probability measures 7→ t + on Φ′ (equipped with the weak topology) is continuous at the origin. Properties of cylindrical and Φ′-valued Lévy processes were studied by the author in [12]. The reader is referred also to [25, 35] for other studies on Lévy and additive processes in the dual of some particular classes of nuclear spaces. Example 3.5. Another important example of semimartingales in the dual of a nuclear space are the solutions to certain stochastic evolution equations. Let Φ be a nuclear Fréchet space and M = (M : t 0)beaΦ′-valued square integrable càdlàg martingale. t ≥ Let A be a continuous linear operator on Φ that is the infinitesimal generator of a (C , 1)-semigroup (S(t) : t 0) of continuous linear operators on Φ (see Definition 1.2 0 ≥ in [19]). Denote by A′ the dual operator of A and by (S′(t) : t 0) the dual semigroup ≥ to (S(t) : t 0). If γ is a Φ′-valued square integrable random variable, it is proved in ≥ Corollary 2.2 in [19] that the homogeneous stochastic evolution equation

′ dXt = A Xtdt + dMt, X0 = γ, has a unique solution X = (X : t 0) given by t ≥ t X = M + S′(t)γ + S′(t s)A′M ds. t t − s Z0 In particular, X is a Φ′-valued càdlàg semimartingale (see Remark 2.1 in [19]). Remark 3.6. Under the additional assumption that Φ′ is a complete nuclear space, another definition of Φ′-valued semimartingales was introduced by Ustünel¨ in [33] by means of the concept of projective system of stochastic process. To explain this concept, let (q : i I) be a family of Hilbertian seminorms generating the nuclear topology i ∈ on Φ′ and let k :Φ′ (Φ′) denotes the canonical inclusion. A projective system i → qi of semimartingales is a family X = (Xi : i I) where each Xi is a (Φ′) -valued ∈ qi semimartingale, and such that if q q then k Xj and Xi are indistinguishable. A i ≤ j qi,qj Φ′-valued processes Y is a semimartingale if k Y = Xi for each i I. It is clear that i ∈ any semimartingale defined in the above sense is also a semimartingale in the sense of Definition 3.3. If Φ is a Fréchet nuclear space the converse is also true as proved by Ustünel¨ in [34] (Theorem II.1 and Corollary II.3). Observe that the definition of Φ′-valued semimartingales as projective systems of semimartingales only makes sense if the strong dual Φ′ is complete and nuclear. The above because if Φ′ does not satisfy these assumptions it might not be possible to express Φ′ as a projective limit of Hilbert spaces with Hilbert-Schmidt embeddings. The following is the main result of this section: Theorem 3.7 (Regularization of cylindrical semimartingales). Let X = (X : t 0) t ≥ be a cylindrical semimartingale in Φ′ such that for each T > 0, the family of linear 0 maps (Xt : t [0, T ]) from Φ into L (Ω, F , P) is equicontinuous (at the origin). Then, ∈ ′ there exists a weaker countably Hilbertian topology θ on Φ and a (Φθ) -valued càdlàg semimartingale Y = (Y : t 0), such that for every φ Φ, Y , φ = ( Y , φ : t 0) t ≥ ∈ h i h t i ≥ is a version of X(φ) = (X (φ) : t 0). Moreover, Y is a Φ′-valued,b regular, càdlàg t ≥ semimartingale that is a version of X and it is unique up to indistinguishable versions. Furthermore, if for each φ Φ the real-valued semimartingale X(φ) is continuous, ∈ ′ ′ then Y is a continuous process in (Φθ) and in Φ .

b 7 Proof. Since for each φ Φ, the real-valued semimartingale (X (φ) : t 0) has a ∈ t ≥ c`adl`ag version (see [7], Section VII.23), the theorem follows using the Regularization Theorem (Theorem 3.2 in [10]).

Corollary 3.8. If Φ is an ultrabornological nuclear space, the conclusion of Theorem 3.7 remains valid if we only assume that each X :Φ L0 (Ω, F , P) is continuous (at t → the origin).

Proof. If Φ is ultrabornological, the continuity of each Xt implies the equicontinuity of the family (X : t [0, T ]) for each T > 0 (see [10], Proposition 3.10). The corollary t ∈ then follows from Theorem 3.7.

Remark 3.9. Let X = (X : t 0) be a cylindrical process in Φ′. The following t ≥ statements are equivalent (see [37], Proposition IV.3.4): (1) For each T > 0, the family of linear maps (X : t [0, T ]) from Φ into L0 (Ω, F , P) t ∈ is equicontinuous (at the origin). (2) For each T > 0, the Fourier transforms of the family (X : t [0, T ]) are equicon- t ∈ tinuous (at the origin) in Φ. Hence the statement of Theorem 3.7 can be formulated in terms of Fourier transforms. D Rd ∞ Rd Example 3.10. Consider the space ( ) := Cc ( ) of test functions and its dual D ′(Rd) the space of distributions. Let z = (z : t 0) be a Rd-valued semimartingale. t ≥ Following [34], we can consider the following cylindrical processes in D ′(Rd): for each t 0, define the map X : D(Rd) L0 (Ω, F , P) by ≥ t → X (φ)= δ (φ)= φ(z ), φ D(Rd), t zt t ∀ ∈ where δ denotes the Dirac measure at x Rd. Each X is continuous from D(Rd) into x ∈ t L0 (Ω, F , P). Moreover, it is a consequence of Itô’s formula that for each φ D(Rd), ∈ (φ(z ) : t 0) is a real-valued semimartingale. Then, (X : t 0) satisfies the condi- t ≥ t ≥ tions in Corolary 3.8, and hence there exists a D ′(Rd)-valued càdlàg semimartingale (Y : t 0) with Radon distributions such that φ D(Rd), P-a.e. Y , φ = φ(z ) for t ≥ ∀ ∈ h t i t all t 0. ≥ Example 3.11. Let X = (X : t 0) is a cylindrical Lévy process in Φ′ such that t ≥ or each T > 0, the family of linear maps (X : t [0, T ]) from Φ into L0 (Ω, F , P) is t ∈ equicontinuous (at the origin). Then its regularized Φ′-valued semimartingale version Y = (Y : t 0) is indeed a Φ′-valued càdlàg Lévy process (see Theorem 3.8 in [12]). t ≥ Moreover, if Φ is a barrelled space, and L = (L : t 0) is a Φ′-valued Lévy process, t ≥ the condition of equicontinuity of the family (L : t [0, T ]) from Φ into L0 (Ω, F , P) t ∈ for all T > 0 is always satisfied (see Corollary 3.11 in [12]).

If X is a Φ′-valued semimartingale satisfying the equicontinuity condition in the statement of Theorem 3.7, then X has a càdlàg semimartingale version. However, as the next result shows there is a large class of examples of nuclear spaces where every Φ′-valued semimartingale always have a càdlàg semimartingale version.

Proposition 3.12. The conclusion of Theorem 3.7 remains valid if Φ is an ultrabornological nuclear space and X = (X : t 0) is a Φ′-valued semimartingale such t ≥ that the probability distribution of each Xt is a Radon measure.

8 Proof. If the probability distribution of Xt is a Radon measure, then by Theorem 2.10 in [10] and the fact that Φ being ultrabornological is also barrelled (see [23], Theorem 11.12.2) imply that each X :Φ L0 (Ω, F , P) is continuous. The result now follows t → from Corollary 3.8. Corollary 3.13. If Φ is a Fréchet nuclear space or the countable inductive limit of Fréchet nuclear spaces, then each Φ′-valued semimartingale X = (X : t 0) possesses t ≥ a càdlàg semimartingale version (unique up to indistinguishable versions). Proof. If Φ is a Fréchet nuclear space or a countable inductive limit of Fréchet nuclear ′ spaces, then every Borel measure on Φβ is a Radon measure (see Corollary 1.3 of Dalecky and Fomin [5], p.11). In particular, for each t 0 the probability distribution ≥ of Xt is Radon. The result now follows from Proposition 3.12. Let X = (X : t 0) be a cylindrical semimartingale in Φ′. Clearly X induces a t ≥ linear map φ X(φ) from Φ into S0. The next result shows that the continuity of 7→ this map is equivalent to the equicontinuity condition in the statement of Theorem 3.7. Proposition 3.14. Let X = (X : t 0) be a cylindrical semimartingale in Φ′. The t ≥ following statements are equivalent: (1) The linear mapping X :Φ S0, φ X(φ), is continuous. → 7→ (2) For each T > 0, the family of linear maps (X : t [0, T ]) from Φ into L0 (Ω, F , P) t ∈ is equicontinuous (at the origin). If any of the above is satisfied, there exists exists a weaker countably Hilbertian topology 0 θ on Φ such that X extends to a continuous map from Φθ into S . Proof. We first prove (1) (2). Observe that because convergence in 0 under the ⇒ S metric d implies uniform convergence in probability onb compact subsets in R+ (see [22], Remarque II.2), then the continuity of the mapping X :Φ S0 implies that → for each T > 0, the family of mappings (X : t [0, T ]) from Φ into L0 (Ω, F , P) is t ∈ equicontinuous (see the proof of Lemma 3.7 in [10]). Now we will prove (2) (1) and the last implication in the statement of Proposition ⇒ ′ 3.14. Let θ be a weaker countably Hilbertian topology on Φ and a (Φθ) -valued càdlàg process Y = (Yt : t 0) as in the conclusion of Theorem 3.7. Because for each φ Φ, ≥ 0 ∈ Y , φ = ( Yt , φ : t 0) is a version of X(φ) = (Xt(φ) : t 0) S b, and because Φ is h i h i ≥ 0 ≥ ∈ dense in Φθ, then Y defines a linear map from Φθ into S . Furthermore, the continuity for each t 0 of Yt on Φθ easily implies that Y is a closed linear map from Φθ into 0 ≥ 0 S . But becauseb Φθ is ultrabornological and Sb is a complete, metrizable, topological vector space, the closedb graph theorem (see [23], Theorem 14.7.3, p.475) impliesb that 0 the map defined byb Y is continuous from Φθ into S . However, because for each φ Φ 0 ∈ we have Y , φ = X(φ) in S , then X extends to a continuous map from Φθ into 0 h i S . Furthermore, because the inclusion fromb Φ into Φθ is linear and continuous, then X :Φ S0 is continuous. b → b If in the proof of Proposition 3.14 we use Corollary 3.8 instead of Theorem 3.7 we obtain the following conclusion: Proposition 3.15. If Φ is an ultrabornological nuclear space, we can replace (2) in Proposition 3.14 by the assumption that each X :Φ L0 (Ω, F , P) is continuous (at t → the origin). Remark 3.16. The implication (1) (2) in Proposition 3.14 remains true if Φ is only ⇒ a locally convex space (see the proof of Lemma 3.7 in [10]).

9 3.2 Regularization of Some Classes of Cylindrical Semimartingales In this section we study how the results obtained in the above section specialize when we restrict our attention to cylindrical local martingales and cylindrical processes of ﬁnite variation. We start with the following regularization result that follows easily from Theorem 3.7.

Proposition 3.17. If in Theorem 3.7 or in Corollary 3.8 the cylindrical process X is a cylindrical local martingale (resp. a cylindrical process of ﬁnite variation), then the regularized version Y of X is a local martingale (resp. a process of ﬁnite variation).

Remark 3.18. If X = (X : t 0) is a cylindrical local martingale (resp. a cylindrical t ≥ process of finite variation) satisfying condition (2) in Proposition 3.14, then it is not true in general that X defines a continuous operator from Φ into the space of real- valued local martingales (resp. of real-valued processes of finite variation ). Mloc V This is a consequence of the fact that and are not closed subspaces of S0 (see Mloc V [9]).

Regularization of cylindrical martingales were studied by the author in [10]. Ob- serve that if the cylindrical martingale has n-moments, for n 2, we obtain a regular- ′ ≥ ized version taking values in some Hilbert space Φq. Theorem 3.19 ([10], Theorem 5.2). Let X = (X : t 0) be a cylindrical martingale t ≥ in Φ′ such that for each t 0 the map X :Φ L0 (Ω, F , P) is continuous. Then, X ≥ t → has a Φ′-valued càdlàg version Y = (Y : t 0). Moreover, we have the following: t ≥ (1) If X is n-th integrable with n 2, then for each T > 0 there exists a continuous ≥ Hilbertian semi-norm q on Φ such that (Y : t [0, T ]) is a Φ′ -valued càdlàg T t ∈ qT martingale satisfying E sup q′ (Y )n < . t∈[0,T ] T t ∞ E n (2) Moreover, if for n 2, supt≥0 ( Xt(φ) ) < for each φ Φ, then there exists ≥ | | ∞ ∈ ′ a continuous Hilbertian semi-norm q on Φ such that Y is a Φq-valued càdlàg martingale satisfying E sup q′(Y )n < . t≥0 t ∞ If for each φ Φ the real-valued process (X (φ) : t 0) has a continuous version, then ∈ t ≥ Y can be chosen to be continuous and such that it satisfies (1) (2) above replacing the − property càdlàg by continuous.

Example 3.20. The following simple example illustrates the usefulness of Theorem 3.19. Let B = (B : t 0) denotes a real-valued Brownian motion. For every t 0 t ≥ ≥ define t X (φ)= φ(s)dB , φ (R). t s ∀ ∈ S Z0 From the properties of the Itôstochastic integral we have that X = (X : t 0) is t ≥ a cylindrical square integrable continuous martingale in the space of tempered dis- ′ 2 2 tributions (R). Moreover, from Itôisometry we have E X (φ) = ½ φ S | t | [0,t] L2(R) φ (R) and since the canonical inclusion from (R) into L2(R) is continuous, it ∀ ∈ S S follows that each X : (R) L0 (Ω, F , P) is also continuous. Theorefore, Theorem t S → 3.19 shows that X has a version Y = (Y : t 0) that is a square integrable continuous t ≥ martingale in the space of tempered distributions ′(R). S

10 We can learn more about X if we analyse further some families of norms on (R). S For every p R, deﬁne on (R) the norm: ∈ S ∞ 2p 2 1 2 φ = n + φ , φ 2 R , || ||p 2 h niL ( ) n=1 X where the sequence of Hermite functions (φn : n = 1, 2,... ) is deﬁned as:

φn+1(x)= g(x) hn(x), n = 0, 1, 2,..., for g(x) = (√2π)−1 exp( x2/2) andp where (h : n = 0, 1, 2,... ) is the sequence of − n Hermite polynomials:

n n ( 1) −1 d hn(x)= − g(x) g(x), n = 0, 1, 2,.... √n! dxn It is known (see e.g. Theorem 1.3.2 in [20]) that the topology in (R) is generated S by the increasing sequence of Hilbertian norms ( p : p = 0, 1,... ). Moreover, if p R ||·|| S denotes the completion of S( ) when equipped with the norm p, then it follows ′ ||·|| that p = −p. S S 2 2 Then since sup E X (φ) = φ 2 R < φ (R), Theorem 3.19(2) shows t≥0 | t | || ||L ( ) ∞ ∀ ∈ S that there exists some p N such that Y is a continuous square integrable martingale ∈ in satisfying E sup Y 2 < . S−p t≥0 || t||−p ∞ In the next result we study regularization for cylindrical processes of locally integrable variation. Observe that in this case one can obtain a regularized version with paths that on a bounded time interval are of bounded variation in some Hilbert space ′ Φρ. We denote by the Banach space of real-valued processes of integrable variation A ∞ a = (a : t 0) equipped with the norm of expected total variation a = E da . t ≥ || ||A 0 | t| Similarly, denotes the linear space of real-valued predictable processes of finite Aloc R variation with locally integrable variation, equipped with the topology of uniform convergence in probability in the total variation on every compact interval [0, T ] of R+. Theorem 3.21. Let A˜ = (A˜ : t 0) be a cylindrical process of locally integrable t ≥ variation, i.e. such that A˜(φ) for each φ Φ. Assume further that for each T > ∈Aloc ∈ 0, the family of linear maps (A˜ : t [0, T ]) from Φ into L0 (Ω, F , P) is equicontinuous t ∈ (at the origin). Then, the cylindrical process A˜ has a Φ′-valued regular càdlàg version A = (A : t 0) (unique up to indistinguishable versions) satisfying that for every t ≥ ω Ω and T > 0, there exists a continuous Hilbertian semi-norm ̺ = ̺(ω, T ) on Φ ∈ such that the map t A (ω) defined on [0, T ] has bounded variation in Φ′ . 7→ t ̺ Proof. First, from Proposition 3.17 there exists a weaker countably Hilbertian topology θ on Φ, and a (Φ )′-valued adapted càdlàg process A = (A : t 0) such that P-a.e. θ t ≥ A , φ = A˜ (φ) t 0, φ Φ. From the above equality it follows that for each φ Φ, h t i t ∀ ≥ ∈ ∈ A , φ . Therefore,b from Proposition 3.14 and since is a closed subspace of h i∈Aloc Aloc S0 (see [22], Théorème IV.7), the process A defines a linear and continuous map from Φ into . We will use the above to show that the paths of A satisfy the bounded θ Aloc variation properties indicated in the statement of the theorem. We will benefit from ideasb taken from [2].

11 Fix T > 0. For every ǫ> 0, using the linearity and continuity of the map A from Φ into , and by following similar arguments to those used in the proof of Lemma θ Aloc 3.3 in [2] there exists a θ-continuous Hilbertian seminorm p on Φ, such that b E i A∆(φ) ,y 2 sup sup 1 e h i ǫ + 2p(φ) , φ Φ, (3.2) ∆ ||y|| ∞ ≤1 −  ≤ ∀ ∈ l∆

  where the sup is taken with respect to an increasing sequence of ﬁnite partitions ∆ of [0, T ] in such a way that the supremum can be attained as a monotone limit. Let us explain the notation used in (3.2). If ∆ = 0 = t < t < < t = T is a ﬁnite { 0 1 · · · n } partition of [0, T ], A∆ denotes the linear and continuous map:

φ A∆(φ) := (A(φ)(t ) A(φ)(t ),...,A(φ)(t ) A(φ)(t )), 7→ 1 − 0 n − n−1 0 1 1 n 1 from Φθ into L (l ), where l denotes the space R equipped with the l -norm y 1 = ∆ ∆ || ||l∆ n Rn Rn ∞ k=1 yk for y = (y1,...,yn) . Recall that in the dual norm to the l -norm | | ∈ n 1 y ∞b = max1≤k≤n yk for y = (y1,...,yn) R is the l -norm, i.e. we have that l∆ ||P|| | | ∈ n x 1 = sup x, y : y ∞ 1 , where , denotes the scalar product in R and l∆ l∆ ||∞|| {|h i| R||n || ≤ } h· ∞·i l∆ denotes the space equipped with the l -norm. Let (p : m N) be an increasing sequence of continuous Hilbertian seminorms on m ∈ Φ generating the topology θ. Since Φ is a nuclear space, we can find and increasing sequence of continuous Hilbertian seminorms (q : m N) on Φ such that m N, m ∈ ∀ ∈ p q , and the inclusion i is Hilbert-Schmidt. We denote by α the countably m ≤ m pm,qm Hilbertian topology on Φ generated by the seminorms (q : m N). By construction, m ∈ the topology α is finer than θ. Let (ǫ : n N) be a sequence of positive real numbers such that ǫ < . n ∈ n n ∞ Then, there exists a subsequence (p : n N) of (p : m N) such that for each mn ∈ m ∈ P n N, ǫ and p satisfy (3.2). To keep the notation simple, we will denote p by ∈ n mn mn pn and the corresponding qmn by qn. Let C > 0 and let ∆ be any member of the increasing sequence of finite partitions of [0, T ] for which the supremum in (3.2) is attained. Let (φqn : j N) Φ be a j ∈ ⊆ complete orthonormal system in Φqn . Then, similar arguments to those used in the proof of Lemma 3.8 in [10] shows that

12 ∆ P sup A (φ) 1 >C l∆ qn(φ)≤1 !

= P sup sup A∆(φ) , y >C  ∞  qn(φ)≤1 ||y||l ≤1 ∆   √e E 1 ∆ 2 1 exp 2 sup sup A (φ) , y ≤ √e 1  − −2C ∞  qn(φ)≤1 ||y||l ≤1 −  ∆   m  √e 1 2 E  ∆ qn = lim sup 1 exp 2 A (φj ) , y m→∞ √e 1  ∞  − −2C  ||y||l ≤1 − ∆  Xj=1 D E    m  √e E  ∆ qn  m lim sup 1 exp i zj A (φj ) , y j=1 NC (dzj), ≤ m→∞ √e 1 Rm ||y|| ∞ ≤1 −    ⊗ Z l∆ j=1 −  X D E   R 2 where NC denotes the centred Gaussian measure on with variance 1/C . Now, from (3.2) the last term can be majorated by 2 m √e qn m lim ǫn + 2pn zjφj j=1 NC (dzj ) m→∞ √e 1 Rm     ⊗ − Z Xj=1  m   2 √e 2 qn lim ǫn + pn φ ≤ m→∞ √e 1  C2 j  − Xj=1 √e 2 2  = ǫn + ipn,qn . √e 1 C2 || ||L2(Φqn ,Φpn ) − From the above bound we have T ∆ P sup dAt(φ) >C = P sup sup A (φ) 1 >C | | l∆ qn(φ)≤1 Z0 ! qn(φ)≤1 ∆ !

∆ = sup P sup A (φ) 1 >C l∆ ∆ qn(φ)≤1 !

√e 2 2 ǫn + ipn,qn , ≤ √e 1 C2 || ||L2(Φqn ,Φpn ) − where, as before, the sup is taken with respect to an increasing sequence of finite partitions of [0, T ] for which the supremum in (3.2) is attained. Then, by taking limit as C we get that →∞ T √e P sup dA (φ) < 1 ǫ . (3.3) | t | ∞ ≥ − √e 1 n qn(φ)≤1 Z0 ! − Now, if for every n N we take ∈ T Λn = sup dAt(φ) < and ΩT = Λn, (qn(φ)≤1 0 | | ∞) N Z N[∈ n\≥N 13 it follows from (3.3), our assumption ǫ < , and the Borel-Cantelli lemma that n n ∞ P(Ω ) = 1. T P Now, recall that the process A was obtained from regularization of the cylindrical process A˜. It is a consequence of the construction of this regularized version (see Remark 3.9 in [10]) that there exists Γ Ω with P(Γ) = 1 such that for all ω Γ, ⊆ ∈ for every t> 0 there exists a θ-continuous Hilbertian semi-norm p = p(ω,t) on Φ such that the map s A (ω) is càdlàg from [0,t] into the Hilbert space Φ′ . 7→ s p If ω Γ Ω , it follows from the construction of the set Ω that there exists ∈ ∩ T T an α-continuous Hilbertian seminorm q = q(ω, T ) on Φ, with p q, and such that ≤ sup T dA (φ)(ω) < . The above clearly implies that φ i′ A(ω) , φ is a q(φ)≤1 0 | t | ∞ 7→ p,q linear and continuous mapping from Φ into the Banach space BV ([0, T ]) of real-valued R q functions that are of bounded variation on [0, T ] equipped with the total variation norm. Let ̺ be a continuous Hilbertian semi-norm on Φ such that q ̺ and iq,̺ is Hilbert- ′ ≤ Schmidt. Then, the dual operator iq,̺ is Hilbert-Schmidt and hence is 1-summing (see [8], Corollary 4.13, p.85). Then, from the Pietsch domination theorem (see [8], Theorem 2.12, p.44) there exists a constant C > 0, and a Radon probability measure ν on the ∗ unit ball Bq (1) of Φq (equipped with the weak topology) such that,

′ ′ ′ ̺ (iq,̺f) C f , φ ν(dφ), f Φq. (3.4) ≤ ∗ |h i| ∀ ∈ ZBq (1) Then, the continuity of the map φ A(ω) i , φ and (3.4) implies that 7→ h ◦ p,q i ̺′ i′ A (ω) i′ A (ω) = ̺′ i′ i′ A (ω) i′ A (ω) p,̺ ti+1 − p,̺ ti q,̺ p,q ti+1 − p,q ti ∆ ∆ X X ′ ′ C ip,qAti+1 (ω) ip,qAti (ω) , φ ν(dφ) ≤ ∗ − ZBq (1) ∆ X T ip,q sup dAt(φ) < . ≤ || ||L2(Φq ,Φp) | | ∞ q(φ)≤1 Z0 The above bound is uniform on ∆, therefore

sup ̺′ i′ A (ω) i′ A (ω) < , p,̺ ti+1 − p,̺ ti ∞ ∆ ∆ X and hence A is a Φ′-valued version of A˜ whose paths satisfy the properties on the statement of the theorem. We ﬁnalize this section by studying regularization of Sp-cylindrical semimartingales. The next results is a generalization of Theorem III.1 in [33].

Theorem 3.22. Let X = (X : t 0) be a Sp-cylindrical semimartingale (1 p< ), t ≥ ≤ ∞ i.e. such that X(φ) Sp for each φ Φ. Assume further that for each T > 0, the ∈ ∈ family of linear maps (X : t [0, T ]) from Φ into L0 (Ω, F , P) is equicontinuous (at t ∈ the origin). Then, there exists a continuous Hilbertian seminorm q on Φ such that X has a càdlàg version Y = (Y : t 0) that is a Sp-semimartingale in Φ′ . Moreover, Y t ≥ q is unique up to indistinguishable versions as a Φ′-valued process. Proof. First, a closed graph theorem argument similar to the one used in the proof of Proposition 3.14 shows that X defines a continuous and linear operator from Φ into

14 Sp. Since Φ is a nuclear space and Sp is Banach, the map X :Φ Sp is nuclear and → then it has a representation (see [31], Theorems III.7.1-2):

∞ X = λ F xi, i i ⊗ Xi=1 where (λ ) l1, (F ) Φ′ equicontinuous, and (xi) Sp bounded. Choose any ǫ> 0, i ∈ i ⊆ ⊆ and for each i a local martingale mi and a process of integrable variation ai such that xi = mi + ai and

∞ i i 1/2 i i [m ,m ]∞ + das < x Sp + ǫ. Z0 Lp(Ω,F ,P)

Now, since (Fi) is equicontinuous and Φ is nuclear, there exists a continuous Hilbertian seminorm q on Φ such that (F ) B (1)0, where B (1)0 denotes the polar set of the i ⊆ q q unit ball Bq(1) of q. Deﬁne

∞ M (ω)= λ F mi(ω), t 0, ω Ω, t i i t ∀ ≥ ∈ Xi=1 and ∞ A (ω)= λ F ai(ω), t 0, ω Ω. t i i t ∀ ≥ ∈ Xi=1 Then, following similar arguments to those used in the proof of Theorem III.1 in [33]) shows that M = (M : t 0) defines a Φ′ -valued càdlàg local martingale with t ≥ q E(sup q′(M )p) < , and that A = (A : t 0) defines a Φ′ -valued càdlàg process t≥0 t ∞ t ≥ q of p-integrable variation. Hence, if we define Y = (Yt : t 0) by Yt = Mt + At t 0, ′ p ≥ ∀ ≥ then Y is is a Φq-valued càdlàg S -semimartingale. Moreover, it is clear from the definition of Y that it is a version of X.

Corollary 3.23. If Φ is an ultrabornological nuclear space, the result in Theorem 3.22 is valid if we only assume that each X :Φ L0 (Ω, F , P) is continuous (at the origin). t →

4 Canonical Representation of Semimartingales in Duals of Nuclear Spaces

Assumption 4.1. Throught this section and unless otherwise speciﬁed Φ will always denote a nuclear space.

The aim of this section is to prove the following theorem that provides a detailed canonical representation for Φ′-valued semimartingales satisfying the equicontinuity condition in the statement of Theorem 3.7. We will show later (see Proposition 4.7) that this equicontinuity condition can be discarded if the nuclear space is ultrabornological, therefore generalizing the canonical representation for semimartingales in the dual Fr´echet nuclear space obtained by P´erez-Abreu in [24] (see Corollary 4.8 below).

Theorem 4.2 (Semimartingale canonical representation). Let X = (X : t 0) be t ≥ Φ′-valued, adapted, c`adl`ag semimartingale such that for each T > 0, the family of linear maps (X : t [0, T ]) from Φ into L0 (Ω, F , P) is equicontinuous (at the origin). t ∈ 15 Given a continuous Hilbertian seminorm ρ on Φ, for each t 0, X admits the unique ≥ t representation

t t c Xt = X0 + At + Mt + fd(µ ν)(s,f)+ fdµ(s,f), (4.1) − c Z0 ZBρ′ (1) Z0 ZBρ′ (1) where (1) X is a -measurable Φ′-valued regular random variable, 0 F0 (2) A = (A : t 0) is a Φ′-valued regular predictable process with uniformly bounded t ≥ jumps satisfying that for every ω Ω and T > 0, there exists a continuous Hilber- ∈ tian semi-norm ̺ = ̺(ω, T ) on Φ such that the map t At(ω) deﬁned on [0, T ] ′ 7→ has bounded variation in Φ̺, c c ′ c (3) M = (Mt : t 0) is a Φ -valued regular continuous local martingale with M0 = 0, ≥ ′ ′ ′ (4) µ(ω; (0,t];Γ) = ½ , Γ (Φ ) with 0 / Γ (here Φ :=Φ 0 ), is 0 0, t { } ≤2 ∀ (c) ′ f , φ 1 ν(ds, df) < , φ Φ, t> 0, 0 Φ |h i| ∧ ∞ ∀ ∈ (5) t fd(µ ν)(s,f) : t 0 ,isa Φ′-valued regular purely discontinuous local 0 RBρ′R(1) − ≥ martingaleR R with uniformly bounded jumps satisfying t t fd(µ ν)(s,f) , φ = f , φ d(µ ν)(s,f), φ Φ, t 0, − h i − ∀ ∈ ≥ *Z0 ZBρ′ (1) + Z0 ZBρ′ (1)

t ′ (6) c fdµ(s,f) : t 0 is a Φ -valued regular adapted càdlàg process which 0 Bρ′ (1) ≥ P hasR R -a.a. paths with only a finite number of jumps on each bounded interval of R+ (in particular is a finite variation process) satisfying

fdµ(s,f)= ∆X ½ c , t 0. s {∆Xs∈Bρ′ (1) } c 0 Bρ′ (1) ∀ ≥ Z Z Xs≤t In order to prove Theorem 4.2 we will need to go through several steps. We benefit from ideas taken from [2] and [24]. First, from Theorem 3.7 there exists a weaker countably Hilbertian topology ϑ on ′ Φ such that X has an indistinguishable version that is a (Φϑ) -valued adapted càdlàg semimartingale. We will identify X with this version. Moreover, from Proposition 3.14 the topology ϑ can be chosen such that X defines a continuousb linear map from Φϑ into S0. Without loss of generality we can assume that ρ is continuous with respect to the topology ϑ. Otherwise, we can just consider another countably Hilbertian topologyb on Φ generated by ρ toguether with the countable family of generating seminorms of ϑ. ′ An important consequence of this remark is that the unit ball Bρ′ (1) of ρ is a bounded ′ subset in (Φϑ) . Now we show the existence of the different components of the representation (4.1). ′ Observe thatb since X is adapted, then X0 is a (Φϑ) -valued random variable that is -measurable. The continuity of the canonical inclusion from (Φ )′ into Φ′ shows that F0 θ X0 satisfies Theorem 4.2(1). b b

16 Now, consider the random measure of the jumps of X:

′ µ(ω; (0,t];Γ) = ½ , t 0, Γ (Φ ). (4.2) {∆Xs(ω)∈Γ} ∀ ≥ ∈B 0 0

∞ ∞ E gdµ = E gdν, (4.3) b ′ b ′ Z0 Z(Φϑ) Z0 Z(Φϑ) and

ν(ω; 0 ; (Φ )′)= ν(ω; R ; 0 ) = 0, { } ϑ + { } ′ ν(ω; t ; (Φϑ) ) 1, t> 0. {b} ≤ ∀ Moreover, since for each φ Φ , X(φ) isb a real-valued semimartingale, then the process ∈ ϑ ( ( ∆X , φ 2 1) : t 0) is locally integrable (see [15], Theorem I.4.47), hence s≤t |h s i| ∧ ≥ from (4.3) we have φ Φ ,bt> 0, P-a.e. P ∀ ∈ ϑ t b f , φ 2 1 ν(ds, df) < . b ′ |h i| ∧ ∞ Z0 Z(Φϑ) ′ ′ Since the canonical inclusion from (Φϑ) into Φ is linear and continuous, the compensator measure ν can be lifted to Φ′ and satisfy (4)(a)-(c) in Theorem 4.2. In a similar way one can show that (see e.g. VIII.68.4b in [7])

t f , φ 2 f , φ ν(ds, df) < φ Φ,t> 0. (4.4) ′ |h i| ∧ |h i| ∞ ∀ ∈ Z0 ZΦ Now, from the properties of µ it is clear that the integral

fdµ(s,f)= ∆X ½ c , t 0, s {∆Xs∈Bρ′ (1) } c ∀ ≥ 0 Bρ′ (1) Z Z Xs≤t ′ is a (Φϑ) -valued adapted càdlàg process which has P-a.a. paths with only a finite number of jumps on each bounded interval of R+. Since the canonical inclusion from ′ b ′ t ′ (Φθ) into Φ is linear and continuous, then c fdµ(s,f) : t 0 asaΦ -valued 0 Bρ′ (1) ≥ process satisfying Theorem 4.2(6). R R b

17 Deﬁne Y = (Y : t 0) by the prescription t ≥ t Yt = Xt X0 fdµ(s,f), t 0. (4.5) − − c ∀ ≥ Z0 ZBρ′ (1)

′ Then, Y is a (Φϑ) -valued adapted c`adl`ag process with Y0 = 0. Moreover, we have the following: b Lemma 4.3. The process Y = (Y : t 0) admits a (unique up to indistinguishable t ≥ versions) representation

Y = M c + M d + A , t 0, (4.6) t t t t ∀ ≥ where M c = (M c : t 0) is a (Φ )′-valued continuous local martingale with M c = 0, t ≥ ϑ 0 M d = (M d : t 0) is a (Φ )′-valued purely discontinuous local martingale and M d = 0, t ≥ ϑ 0 A = (A : t 0) is a (Φ )′-valuedb predictable càdlàg process of locally integrable t ≥ ϑ variation and A0 = 0. b b Proof. First, observe that by construction the set of jumps ∆Yt : t 0 of Y is ′ { ≥ } contained in the bounded subset Bρ′ (1) in (Φϑ) , and consequently ∆Yt : t 0 is ′ { ≥ } itself bounded in (Φϑ) . Therefore, the definition of strong boundedness implies that for every bounded subset C in Φϑ there existsb a KC > 0 such that b supb sup ∆Yt , φ < KC . (4.7) φ∈C t≥0 |h i|

However, since for each φ Φ the set φ is bounded in Φ , then (4.7) shows that the ∈ ϑ { } ϑ real-valued semimartingale Y , φ has uniformly bounded jumps, hence is a special h i semimartingale and has a (unique)b representation (see theb proof of Theorem III.35 in [28], p.131) Y , φ = M φ + Aφ, t 0, h t i t t ∀ ≥ φ φ where (Mt : t 0) is a real-valued càdlàg local martingale, (At : t 0) is a real- ≥ ≥φ φ valued predictable càdlàg process of locally integrable variation, and M0 = A0 = 0. Furthermore, each M φ has a (unique) representation (see [7], Theorem VIII.43, p.353)

M φ = M c,φ + M d,φ, t 0, t t t ∀ ≥ where (M c,φ : t 0) is a real-valued continuous local martingale, (M d,φ : t 0) is t ≥ t ≥ a real-valued purely discontinuous local martingale; these two local martingales are orthogonal. Therefore, Y , φ has the (unique) representation h i Y , φ = M c,φ + M d,φ + Aφ, t 0. (4.8) h t i t t t ∀ ≥ ˜ c 0 F P ˜ c If for every t 0 we define the mappings Mt : Φϑ L (Ω, , ), φ Mt (φ) := c,φ ˜ d ≥ 0 F P ˜ d d,φ→ ˜ 7→0 F P Mt , Mt : Φϑ L (Ω, , ), φ Mt (φ) := Mt , and At : Φϑ L (Ω, , ), ˜ →φ 7→ b → φ At(φ) := At , then by uniqueness of the decomposition (see the arguments in 7→ ˜ c ˜ d ˜ the proof ofb Theorem 3.1 in [2]) the maps Mt , Mt and At areb linear. Therefore, ˜ c ˜ c ˜ d ˜ d ˜ ˜ M = (Mt : t 0), M = (Mt : t 0) and A = (At : t 0) are cylindrical ≥ ′ ≥ ≥ 0 semimartingales in (Φϑ) and hence define linear maps from Φϑ into S . Our next objective is to show they are also continuous. Since Φϑ is ultrabornological, it is b b 18 b enough to show that the maps M˜ c, M˜ d and A˜ are sequentially closed, because in that case the closed graph theorem shows that they are continuous ([23], Theorem 14.7.3, p.475). Let φ φ in Φ and suppose that M˜ c(φ ) mc, M˜ d(φ ) md and A˜(φ ) a n → ϑ n → n → n → in S0. By the continuity of Y from Φ into S0, we have Y (φ ) Y (φ) in S0. Since ϑ n → the set C = φ : nb N is bounded in Φ , then from (4.7) the family of real-valued { n ∈ } θ semimartingales Y , φ : n N bhas jumps uniformly bounded by K . But as {h ni ∈ } C the collection of all the real-valued semimartingalesb with jumps uniformly bounded by K is a closed subspace in S0 (see [22], Theorem IV.4), then Y , φ also has jumps C h i uniformly bounded by K . Therefore, because Y (φ ) Y (φ) in S0 we have that (see C n → Remarque IV.3 in [22]) M˜ c(φ ) M˜ c(φ), M˜ d(φ ) M˜ d(φ) and A˜(φ ) A˜(φ) in S0. n → n → n → Hence, by uniqueness of limits we get that mc = M˜ c(φ), md = M˜ d(φ) and a = A˜(φ). c d 0 Thus, the mappings M˜ , M˜ and A˜ are continuous from Φϑ into S . Hence, from Propositions 3.14 and 3.17, there exists another weaker countably ′ Hilbertian topology θ on Φ, larger than ϑ, and a (Φθ) -valuedb continuous local martingale M c = (M c : t 0), a (Φ )′-valued purely discontinuous local martingale t ≥ θ M d = (M d : t 0), and a (Φ )′-valued predictable càdlàgb process of locally integrable t ≥ θ variation A = (A : t 0) with A =b 0, all of them such that P-a.e. t 0, φ Φ, t ≥ 0 ∀ ≥ ∈ b M c , φ = M˜ c(φ), (4.9) h t i t d ˜ d Mt , φ = Mt (φ), (4.10) D A , φE= A˜ (φ). (4.11) h t i t But then, (4.8), (4.9), (4.10) and (4.11) imply that P-a.e. t 0, φ Φ, ∀ ≥ ∈ Y , φ = M c , φ + M d , φ + A , φ . (4.12) h t i h t i t h t i D E Now, since the processes Y , M c, M d and A are regular processes, then (4.12) together with Proposition 2.12 in [10] implies that Y , M c, M d and A satisfy (4.6).

Lemma 4.4. The processes M d = (M d : t 0) and A = (A : t 0) defined in t ≥ t ≥ Lemma 4.3 have P-a.e. uniformly bounded jumps in Φθ. Proof. Let φ Φ . By the definition of local martingale and of process of integrable ∈ θ variation, there exists a sequence of stopping times (τb : n N) increasing to such n ∈ ∞ that for all n Nb, ( M c , φ : t 0) and ( M d , φ : t 0) are uniformly inte- ∈ t∧τn ≥ t∧τn ≥ grable martingales and ( A , φ : t 0) is of integrable total variation. Moreover, h t∧τn i ≥ since ( A , φ : t 0) is predictable and ( M c , φ : t 0) is continuous, it then follows h t i ≥ h t i ≥ from (4.12) that P-a.e. t 0, ∀ ≥ d E ( ∆Y , φ − )= E ∆M , φ + ∆A , φ − = ∆A , φ . (4.13) h t∧τn i Ft t∧τn h t∧τn i |Ft h t∧τn i D E ′ Now, recall that the set of jumps ∆Yt : t 0 of Y is bounded in (Φϑ) , but since the { ≥ } ′ ′ topology θ is finer than ϑ the canonical inclusion from (Φϑ) into (Φθ) is continuous; then ∆Y : t 0 is bounded in (Φ )′. Hence, from (4.7) with C b= φ and taking { t ≥ } θ { } limits as n in (4.13), we have that P-a.e. t 0, b b →∞ ∀ ≥ b E ( ∆Y , φ − )= ∆A , φ . h t i |Ft h t i

19 Therefore, for any given bounded subset C in Φθ (recall that this space is separable), it follows from (4.7) that P-a.e. b sup sup ∆At , φ < KC . t≥0 φ∈C |h i|

But the above inequality together with (4.12) shows that P-a.e.

d sup sup ∆Mt , φ < 2KC . t≥0 φ∈C D E

d Hence, the processes M and A have P-a.e. uniformly bounded jumps in Φθ. Lemma 4.5. For each t> 0 and φ Φ, ∈ b t d Mt , φ = f , φ d(µ ν)(s,f). (4.14) 0 B ′ (1) h i − D E Z Z ρ Proof. First, for each φ Φ it is a consequence of (4.4) (see Theorem 2.1 in [18]) ∈ that t f , φ d(µ ν)(s,f) : t 0 is a purely discontinuous local martingale 0 Bρ′ (1) h i − ≥ satisfyingR R t ∆ f , φ d(µ ν)(s,f) = f , φ µ( t ,f) f , φ ν( t ,f). h i − h i { } − h i { } Z0 ZBρ′ (1) ! ZBρ′ (1) ZBρ′ (1) Thus, since ( M d , φ : t 0) is also a purely discontinuous local martingale, to prove t ≥ (4.14) it is enough to show that

d ∆ Mt , φ = f , φ µ( t ,f) f , φ ν( t ,f). B ′ (1) h i { } − B ′ (1) h i { } D E Z ρ Z ρ But the above equality follows from exactly the same arguments to those used in the proof of Theorem 3 in [24], so we leave the details to the reader. To finallize the proof of Theorem 4.2, observe that since the canonical inclusion from ′ ′ c d ′ (Φθ) into Φ is linear and continuous, then A, M and M define Φ -valued processes. From Lemma 4.3 we have that M c satisfies Theorem 4.2(3). For A, it follows from Lem- masb 4.3 and 4.4, and Theorem 3.21, that A has an indistinguishable version that satis- d t fies Theorem 4.2(2). If for each t 0 we denote Mt by ′ fd(µ ν)(s,f), then ≥ 0 Bρ (1) − it is a consequence of Lemmas 4.3, 4.4 and 4.5 that t R R fd(µ ν)(s,f) : t 0 0 Bρ′ (1) − ≥ satisfies Theorem 4.2(5). Finally, the fact that XRadmitsR the representation (4.1) follows from (4.5) and (4.6).

′ Remark 4.6. In the proof of Theorem 4.2 we have used a (Φϑ) -valued indistinguishable version of X, which at a ﬁrst glance might lead us to conclude that the random measure µ of the jumps of X depends on the countably Hilbertianb topology ϑ on Φ. However, this is not the case as the aforementioned version of X is unique (up to indistinguishable versions) as a Φ′-valued process (see Theorem 3.7). Therefore, the random measure µ of the jumps of X deﬁned in (4.2) is unique and consequently its compensated measure ν. Hence µ and ν do not depend on the given seminorm ρ. Sim- ilarly, the continuous local martingale part M c is independent of the given seminorm

20 ρ. To prove this, suppose M 1,c and M 2,c denote the continuous local martingale part of any two representations of X as in (4.1). For any given φ Φ, let Xc,φ denotes the ∈ real-valued continuous local martingale corresponding to the canonical representation of X , φ . Then, by uniqueness of Xc,φ we have P-a.e. h i M 1,c , φ = Xc,φ = M 2,c , φ , t 0. t t t ∀ ≥ D E D E But since M 1,c and M 2,c are regular processes, Proposition 2.12 in [10] shows that M 1,c = M 2,c. Under additional assumptions on Φ, the next result shows that the representation (4.1) remains valid for any Φ′-valued semimartingale: Proposition 4.7. The conclusion of Theorem 4.2 remains valid if Φ is an ultrabornological nuclear space and X = (X : t 0) is a Φ′-valued càdlàg semimartingale t ≥ such that the probability distribution of each Xt is a Radon measure. Proof. The result follows if in the proof of Theorem 4.2 we use Corollary 3.8 instead of Theorem 3.7 and if instead of Proposition 3.14 we use Corollary 3.15. Corollary 4.8. If Φ is a Fréchet nuclear space or the countable inductive limit of Fréchet nuclear spaces, then each Φ′-valued càdlàg semimartingale X = (X : t 0) t ≥ possesses a representation satisfying the conditions in Theorem 4.2. Proof. When Φ is a Fréchet nuclear space or a countable inductive limit of Fréchet ′ nuclear spaces, then every Borel measure on Φβ is a Radon measure (see Corollary 1.3 of Dalecky and Fomin [5], p.11). In particular, for each t 0 the probability ≥ distribution of Xt is Radon. The result now follows from Proposition 4.7. The representation in Theorem 4.2 leads naturaly to the following definition: Definition 4.9. Let X = (X : t 0) be Φ′-valued adapted càdlàg semimartingale t ≥ such that for each T > 0, the family of linear maps (X : t [0, T ]) from Φ into t ∈ L0 (Ω, F , P) is equicontinuous (at the origin). Given a continuous Hilbertian seminorm ρ on Φ, we call characteristics of X relative to ρ the triplet (A,C,ν) consisting in: (1) A is the Φ′-valued process defined in Theorem 4.2(2). (2) C :Ω R Φ Φ is the map defined by × + × × C(ω,t,φ,ϕ)= M c , φ , M c , ϕ (ω), (ω,t,φ,ϕ) Ω R Φ Φ, hhh i h i iit ∀ ∈ × + × × where M c is as given in Theorem 4.2(3), and M c , φ , M c , ϕ is the angle hhh i h i ii bracket process of the continuous local martingales M c , φ and M c , ϕ . h i h i (3) ν is the (predictable) compensator measure of the random measure µ associated to the jumps of X, as given in Theorem 4.2(4). Remark 4.10. It follows from Remark 4.6 that C and ν do not depend on the choice of the seminorm ρ on Φ, while A = A(ρ) does.

5 Characteristics and L´evy Processes

In this section we study in detail the canonical decomposition and characteristics of a Φ′-valued Lévy process and how they relate with their Lévy-Itôdecomposition studied in [12].

21 Let L = (L : t 0)beaΦ′-valued càdlàg Lévy process. Suppose that for every T > t ≥ 0, the family (L : t [0, T ]) of linear maps from Φ into L0 (Ω, F , P) is equicontinuous t ∈ (at the origin) (see Example 3.11). Under the above conditions it is proved in Section 4 in [12] that the random measure µ of the jumps of L is a Poisson Random measure, that we denote by N, and that its (predictable) compensator measure is of the form ν(ω; dt; df)= dtν(df), where ν is a Lévy measure on Φ′ in the following sense (see [12], Theorem 4.11): (1) ν( 0 ) = 0, { } (2) for each neighborhood of zero U Φ′, the restriction ν of ν on the set U c ⊆ U c belongs to the space Mb (Φ′) of bounded Radon measures on Φ′, R (3) there exists a continuous Hilbertian semi-norm ρ on Φ such that

′ 2 Mb ′ ρ (f) ν(df) < , and ν c R(Φ ), (5.1) ∞ Bρ′ (1) ∈ ZBρ′ (1)

′ 0 where recall that ρ is the dual norm of ρ (see Sect. 2) and Bρ′ (1) := Bρ(1) = f Φ′ : ρ′(f) 1 . { ∈ ≤ } Here is important to stress the fact that the seminorm ρ satisfying (5.1) is not unique. Indeed, any continuous Hilbert semi-norm q on such that ρ q satisﬁes (5.1). ≤ It is shown in Theorem 4.17 in [12] that relative to a continuous Hilbertian seminorm ρ on Φ satisfying (5.1), for each t 0, L admits the unique representation ≥ t

Lt = tm + Wt + fN(t, df)+ fN(t, df) (5.2) c ZBρ′ (1) ZBρ′ (1) that is usually called the Lévy-Itôdecompositione of L. In (5.2), we have that m Φ′, ∈ N(dt, df) = N(dt, df) dt ν(df) is the compensated Poisson random measure, and − (W : t 0) is a Φ′-valued Lévy process with continuous paths (also called a Φ′-valued t ≥ Wienere process) with zero-mean (i.e. E( W , φ ) = 0 for each t 0 and φ Φ) and h t i ≥ ∈ covariance functional satisfying Q E ( W , φ W , ϕ ) = (t s) (φ, ϕ), φ, ϕ Φ, s,t 0. (5.3) h t i h s i ∧ Q ∀ ∈ ≥ Observe that is a continuous, symmetric, non-negative bilinear form on Φ Φ. It is Q × important to remark that all the random components of the representation (5.2) are independent. Observe that when we compare (5.2) with (4.1), we conclude that At = tm, c t t Mt = Wt, and that ′ fd(µ ν)(s,f) and ′ c fdµ(s,f) corresponds to 0 Bρ (1) − 0 Bρ (1) the Poisson integrals fN(t, df) and c fN(t, df) (for details on the defini- R BRρ′ (1) Bρ′ (1) R R tion of Poisson integrals see [12]). Moreover, since for each φ Φ we have that W , φ R R ∈ h i is a real-valued Wiener process,e then (see Theorem II.4.4 in [15]) it follows from (5.3) that C(ω,t,φ,ϕ)= W , φ , W , ϕ (ω)= E ( W , φ W , ϕ )= t (φ, ϕ), hhh i h i iit h t i h t i Q for all (ω,t,φ,ϕ) Ω R Φ Φ. Therefore, we conclude that the characteristics ∈ × + × × of L relative to ρ satisfying (5.1) are deterministic and have the form: A = tm, C(ω,t,φ,ϕ)= t (φ, ϕ), ν(ω; dt; df)= dt ν(df). (5.4) t Q The next result shows that the above form of the characteristics is exclusive of the Φ′-valued Lévy processes:

22 Theorem 5.1. Let L = (L : t 0) be Φ′-valued, adapted, càdlàg process such that t ≥ for each T > 0, the family of linear maps (L : t [0, T ]) from Φ into L0 (Ω, F , P) is t ∈ equicontinuous (at the origin). Then, L is a Φ′-valued Lévy process if and only if it is a Φ′-valued semimartingale and there exists a continuous Hilbertian semi-norm ρ on Φ such that the characteristics of L relative to ρ are of the form (5.4), where m Φ′, ∈ is a continuous, symmetric, non-negative bilinear form on Φ Φ, and ν is a Lévy Q × measure on Φ for which ρ satisfy (5.1). Moreover, for all t 0, φ Φ, ≥ ∈ E eihLt ,φi = etη(φ), with (5.5) 1 ihf,φi

η(φ)= i m , φ (φ, φ)+ e 1 i f , φ ½ (f) ν(df). Bρ′ (1) h i− 2Q Φ′ − − h i Z Proof. We have already proved that if L is a Φ′-valued Lévy process then its characteristics relative to ρ are of the form (5.4). Moreover, in that case (5.5) corresponds to its Lévy-Khintchine formula (see Theorem 4.18 in [12]). Assume now that L is a Φ′-valued semimartingale with canonical representation (4.1) and such that the characteristics of L with respect to ρ are of the form (5.4). We will show that L induces a cylindrical Lévy process in Φ′. Let φ ,...,φ Φ and let L(φ ,...,φ ) = (L (φ ,...,φ ) : t 0) defined by 1 n ∈ 1 n t 1 n ≥ L (φ ,...,φ ) := ( L , φ ,..., L , φ ), t 0. t 1 n h t 1i h t ni ∀ ≥ n From the corresponding properties of L, it is clear that L(φ1,...,φn) is a R -valued càdlàg adapted semimartingale. We now study the form of its characteristics (a, c, λ)

(Deﬁnition II.2.6 in [15]) relative to the truncation function h(x) = x½B, where B = π (B ′ (1)) and π :Φ′ Rn is the projection φ1,...,φn ρ φ1,...,φn → π (f) = ( f , φ ,..., f , φ ), f Φ′. φ1,...,φn h 1i h ni ∀ ∈ c We study ﬁrst c = (cij)i,j≤n. Let M (φ1,...,φn) denotes the continuous local n martingale part of the R -valued process L(φ1,...,φn). From (4.1) and the uniqueness of the continuous martingale part it follows that:

M c(φ ,...,φ ) = ( M c , φ ,..., M c , φ ), t 0. (5.6) t 1 n h t 1i h t ni ∀ ≥

Then, from the deﬁnition of c = (cij)i,j≤n, (5.4) and the corresponding properties of , for each i, j n we have Q ≤ c (t,ω)= M c , φ , M c , φ (ω)= tq , ω Ω, t 0, (5.7) i,j hhh ii h ji iit i,j ∀ ∈ ≥ where q = (q ) , deﬁned by q = (φ , φ ), is a symmetric non-negative n n ij i,j≤n i,j Q i j × matrix. Now, λ is the predictable compensator measure of the random measure ξ of the jumps of L(φ ,...,φ ). For a set D (Rn) (Rn = Rn 0 ), we have that 1 n ∈B 0 0 \ { }

ξ(ω; (0,t]; D) = ½D (∆Ls(φ1,...,φn)(ω)) 0≤Xs≤t

½ − = π 1 (D) (∆Ls(ω)) φ1,...,φn 0≤Xs≤t = µ(ω; (0,t],π−1 (D)). (5.8) φ1,...,φn

23 But since the predictable compensated measure is unique, we have that for each D ∈ (Rn), B 0 λ(ω; t; D)= ν(ω; t; π−1 (D)). (5.9) φ1,...,φn Then, from (5.4) we have that

λ(ω; dt; dy)= dt ν π−1 (dy). (5.10) ◦ φ1,...,φn Moreover, ν π−1 is a L´evy measure on Rn since Theorem 4.2(4)(c) implies that ◦ φ1,...,φn t t y 2 1 ν π−1 (dy) = y 2 1 ds ν π−1 (dy) φ1,...,φn φ1,...,φn Rn | | ∧ ◦ 0 Rn | | ∧ ◦ Z Z t Z 2 = πφ1,...,φn (f) 1 ν(ds, df) < . ′ | | ∧ ∞ Z0 ZΦ We now study the Rn-valued predictable ﬁnite variation process a = (a : t 0). First, t ≥ observe that (5.8) and (5.9) show that for all t 0, ≥ t t f d(µ ν)(s,f) , φ ,..., f d(µ ν)(s,f) , φ − 1 − n *Z0 ZBρ′ (1) + *Z0 ZBρ′ (1) +! t = y d(ξ λ)(s,y), (5.11) − Z0 ZB and

t t f dµ(s,f) , φ1 ,..., f dµ(s,f) , φn c c *Z0 ZBρ′ (1) + *Z0 ZBρ′ (1) +! t = y dξ(s,f). (5.12) c Z0 ZB

Then, by considering the canonical decomposition of L(φ1,...,φn), (4.1), (5.4), (5.6), (5.11), and (5.12), it follows that

a (ω) = ( A (ω) , φ ,..., A (ω) , φ )= tm, ω Ω, t 0, (5.13) t h t 1i h t ni ∀ ∈ ≥ where m = ( m , φ ,..., m , φ ) Rn. h 1i h ni ∈ Now, the particular form of the characteristics (a, c, λ) of the Rn-valued semimartingale L(φ1,...,φn) given in (5.13), (5.7), (5.10) respectively, and from Corollary II.4.19 d in [15], it follows that L(φ1,...,φn) is a R -valued Lévy process. Therefore, L induces a cylindrical Lévy process in Φ′. But then, Theorem 3.8 in [12] shows that L has an indistinguishable version that is a Φ′-valued Lévy process. Hence, L is itself a Φ′-valued Lévy process.

Acknowledgements The author acknowledge The University of Costa Rica for providing financial support through the grant “B9131-Análisis Estocástico con Procesos Cil´ındricos”. Many thanks are also due to the referees for their helpful and insightful comments that contributed greatly to improve the presentation of this manuscript.

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